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Sequences and Series of Functions
Published in John Srdjan Petrovic, Advanced Calculus, 2020
Theorem 8.4.3. established that a power series always converges in an interval of radius R. We call them the interval of convergence and the radius of convergence. Still, some questions remain open. What about the endpoints of the interval? In Example 8.4.2, R = 1, so we are interested in x = 1 and x = —1. In the former case we have the series Σn=0∞ 1/n2 which converges as a p-series; in the latter case, we have Σn=0∞(-1)n/n2 which converges absolutely. Conclusion: the series converges for both x —R and x = R. Before we start believing that this is always true, let us look at a few more examples.
Infinite series
Published in C.W. Evans, Engineering Mathematics, 2019
R is known as the radius of convergence of the power series. Every power series in x converges when x = 0, and if this is the only value of x for which it converges we say it has zero radius of convergence and write R = 0. Some power series in x converge for all x, and we then say the power series has an infinite radius of convergence and write R ∞.
Series Solutions: Preliminaries (A Brief Review of Infinite Series, Power Series and a Little Complex Variables)
Published in Kenneth B. Howell, Ordinary Differential Equations, 2019
The radius of convergence for a given power series can sometimes be determined through careful use of the formulas in either the limit ratio test or the limit root test. You may recall doing this. We, however, will discover that the radii of convergence for the power series of interest to us can be determined much more easily from the “singularities” of whatever differential equation we will be trying to solve.
A novel weighted local averaging for the Galerkin method with application to elastic buckling of Euler column
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2022
Anh Tay Nguyen, Nguyen Cao Thang, Tran Tuan Long, N. D. Anh, P. M. Thang, Nguyen Xuan Thanh
The problem of buckling of structures has been investigated by many methods, such as, Energy method [25], Variational iteration method [26], [27], Homotopy perturbation method [28], [29] and Differential quadrature method [30]. A novel method for bending and buckling analysis of simply supported rectangular nano plates has been proposed by Mousavi et al. [31]. A complete representation of strong-form of governing equation and boundary conditions are derived based on the infinite series of modified nonlocal constitutive equations. A radius of convergence for the computed series is calculated to make sure the analyses are valid and reliable. Research on buckling of columns has been the center point of study for many researchers and become more systematic during the last decades.